Expected-Value-of-a-Sample-Estimate-Pages---Yield.pdf

About Global Documents

Global Documents provides you with documents from around the globe on a variety of topics for your enjoyment.
Global Documents utilizes edocr for all its document needs due to edocr's wonderful content features. Thousands of professionals and businesses around the globe publish marketing, sales, operations, customer service and financial documents making it easier for prospects and customers to find content.
sample estiM error variance element simple random sample
C1 C1
Uw
1\
:,1
\
I,
95
Madow. ~/
The conditions are usually satisfied with regard to estimates
from sample surveys.
As a rule of thumb the variance formula is usually
accepted
'as satisfactory if the coefficient of variation of the variable
o
w
in the denominator is less than 0.1; that is, if -- < 0.1.
In other words,
w
this condition states that the coefficient of variation of the estimate in
the denominator should be less than 10 percent.
A larger coefficient of
variation might be tolerable before becoming concerned about Equation (3.26)
as an approximation.
o
The condition ~
< 0.1 is more strin~ent than necessary for reg~rding
w
the bias of a ratio as negligible.
With few exceptions in practice the
bias of a ratio is ignored.
Some of the logic for this ,,,illappear in
the illustration below.
To summarize, the conditions when Equations (3.25)
and (3.26) are not gpod approximations are such that the ratio is likely to
be of questionable value owing to large variance.
If u and ware
linear combinations of random variables, the theory
presented in previous sections applies to u and to w. Assumin~ u and w
are estimates from a sample,
u take into
the
to estimate Var(-)
account
w
sample design and substitute in Equation (3.26) estimates of u, W,
2
2
0u' o ,
w
and p
• Ignore Equation (3.25) unless there is reason to believe the bias
uw
of the ratio mi~ht be important relative to its standard error.
It is of interest to note the similairity between Var(u-w) and var<;).
According to Theorem 3.5,
Var(u-w) g 02 + 02 - 2p
u
w
uw
JJ Hansen, Hurwitz, and Hadow, Sample Survey Methods and Theory,
Volume I, Chapter 4, John Wiley and Sons, 1953.
96
By definition the relative variance of ~n estimate is the variance of the
estimate divided by the s~uare of its expected value.
Thus, in terms of
7
the relativ~ variance of a ratio, Equation (3.26) can be written
2
2
a
a
ReI Var(~) ••~ + ~ - 2p
w
-2
-2
uw
u
w
a au ,,,
uw
The similarity is an aid to remembering the formula for Var(u).
w
elements from a population of N. Let x and y be the sample means for
Illustration 3.13. Suppose one has a simple random sample of n
,.
r
characteris tics X and Y. Then. u - x. w ••• Y.
2
a
u
N-n
.. --
N
n
and
2
a .••
w
N-n
N
n
N-n
N
Notice that the condition discussed above,
sample is large enough so
2
Sy
2
---=2 < 0.1
nY
a
w
< 0.1. is satisfied if the
w
Substituting in Equation (3.26) we obtain the following as the variance of
the ratio:
x2 s~ si
-[-+--
-2 -2
-2
Y
X
Y
2PXYSXSy
---]
XY
For this illustration it becomes
y
Equation (3.25).
The bias of ~
X
as an estimate of -
y
is given by the second term of
S2
Y
[- -
-2
y .
As the size of the sample increases. the bias decreases as l whereas the
n
standard error of the ratio decreases at a slower rate. namely
1
10
\
\
.'),,;
,r
•
'f ~,
j
9.7
Thus, we need not be concerned about a possibility of the bias becominr,
important relative to samplinr, error as the size of the sample increases.
A possible exception occurs when several ratios are combined.
An example
is stratified random sampling when,many strata are involved and separate
ratio estimates are made for the strata.
This
is discussed in the books
on sampling.
3.9 CONDITIONAL EXPECTATION
The theory for conditional expectation and conditional variance of a
random variable is a very important part of sampling theory, especially
,II
::
in the theory for multistage sampling.
The theory will be discussed with
reference to two-stage sampling.
,
•..I.~
I
,_
The notation that will be used in this and the next section is as
fQllows:
M is the number of psu's (primary sampling units) in the population.
m is the number of psu's in the sample.
th
Ni is the total number of elements in the i
psu.
,
"
~i,~~~~,,:
'\
M
N ••~Ni is the total number of elements in the population.
i
th
ni is the sample number of elements from the i
psu.
m
n - Eni is the total number of elements in the sample.
i
n
n--m
Xij is the value of X for the jth element In the i
th osu.
It
refers to an element in the population, that is, j ••1,••• , Ni,
and i-I,
•••, M.
••.••,••I
:;"11' "l.
~ "':r....
..,.'
~~.:....
"r·.~,r'-i~~.:
.
';Z',:",',,'
"'1: ~',.~~
,
,
. ,~
98
Xij is the value of X for the jth element in the sample from the
th
i
psu in the samplet that iSt the indexes i and j refer to
the.set of psu's and elements in the sample.
Ni
~h
X
- L X
is the populati~n total for thei
psu.
i·
j
ij
Xi.
th
Xi. - --
is the average of X for all elements in the i
psu.
Ni
MNi
M
1:1: X
ij
LXi.
X
ij
i
is the average of all N elements.
-
- --
..
N
N
, I
I r ~Ii
,
'
!
I
,
L
/
i
is Ithe ~verage of the psu totals.
'",
'
,
,
"
difference between X •• and X.
Be sure to note the
, '
Xi_ - ~iXij ri~i~he~ampleltotal :for the ith psu in the sa~le.
j
Xi.:
xi. = n
i
is the average for the ni elements in the sample from
th
the i
psu.
" .
~.
"
X
n
is the average for all elements in the sample.
Assume simple random samplingt equal probability of selection without
replacementt at both stages. Consider the saMple of ni elements from the
ili
-
i
psu. We know from Section 3.3 that xi' is an unbiased estimate of the
psu mean Xi. ; that iSt E(Xi.) = Xi. and for a fixed i (a specified psu)
ENixi• - NiE(Xi.) - NiXi• - Xi ••
Butt owing to the first stage of samplinP,t
99
ENixi must be treated as a random variable. Hence, it is necessary to
become involved with the expected value of an expected value.
First, consider X as a random variable, in the context of single-
equal anyone of .the values Xii in the
Let P(ij) be the probability of selecting
being equal to Xij• By definition
MNi
E(X) = &E P(ij)Xij
ij
stage sampline, which could
M
population set of N = ENi •
i
th
th
the j
element in the i
psu; that is, P(ij) is the probability of X
(3.27)
: " .:' l,',~
:1, r ,I ',I,~,
i: f 11:1
~ r,'
q.l'
'
Now consider the selection of an element as a two-step procedure:
••
I •~
...,.
"
'M i !' l
It'
:,
~.,
(I) selected a psu with probability P(i), and (2) selected an element
i
within the selected psu ldth prob'abilitypol i)~ In words,P(j Ii) is the
th
th
probability of selecting the j
element in the i
p~u given that the
.th
hId
bId
1
psu as a rea y een se ecte •
stitutio~,Equation (3.27) becomes
Thus, P(ij) •••,P(i)p(jli). By sub-
MN
E(X) = ~~ip(i)P(jli)Xij
"
.
".,
.:;'.... ,.
..
or
E(X)
(3.28)
By defini tion ,
N
i& P(j1i)Xij is the expected value of X for a fixed value
j
of 1. It is called"conditional expectation. II
N.
Let E2(Xli) = ~1p(jli)Xij where E2(xli) is the form of notation we
will be using to designate conditional expectation.
To repeat, E2(xli)
means the expected value of X for a fixed i. The subscript 2 indicates
100
that the conditional expectation applies to the second sta~e of sampling.
El and EZ will refer to expectation at the first and second stages,
respectively.
the expected value
In fact E2(xli)
is P(i).
Thus the
Substituting Ez(Xli) in Equation (3.Z8) we obtain
M
E(X) = EP(i) E2(xli)
i
There is one value of Ez(Xli) for each of the M psu's.
is a random varigble where the probability of Ez(Xli)
right-hand side of Equation (3.Z9) is, by definition,
of Ez(Xli).
This leads to the followin~ theorem:
Theo rem 3.6. E(X) - ElE2(xli)
------
Suppose P (j Ii) 1
and P(i)
I
Then,
--
- -
Ni
M·
I
(3.Z9)
,I
Ii
j
I
'i
and
In this case E(X) is an unweighted average of the psu averages.
It is
.,:;
important to note that,if P(i) and P(jli) are chosen in such a way that
P(ij) is constant, every element has the same chance of selection.
This
point will be discussed later.
Theorem 3.3 dealt wi titthe expected value of a linear combination of
random variables.
There is a corresponding theorem for conditional expecta-
tion.
Assume the linear combination is
k
U - alul+···+~~
- t:latut
';,' ..,
'~ "~
J,
)
ii.
l~ r
-'I
;,
I"
I"
,
,
,
\
101
where al""'~
are constants and ul"",uk
are random variables.
Let
E(Ulci) be the expected v~lue of U under a specified conditio~ ci' where
ci is one'of the conditions out of a set of M conditions that could occur.
The theorem on conditional expectation can then be stated symbolically as
follows:
Compare Theorems 3.7 and 3.3 and note that Theorem
3.7 is like
Theorem 3.3 except that conditional expectation is applied.
Assume c is
a random event and that the probability of the event ci occurring is p(i1.
Then E(U/Ci) is a random variable and by definition the expected value of
M
E(ulci) is EP(i)E(ulci) which is E(U).
Thus, we have the fo1lowin~
i
theorem:
Theorem 3.8. The expected value of U is the expected value of the
conditional expected value of U, which in symbols is written as follows:
(3.30)
Substituting the value of E(U/ci) from Theorem 3.7 in Equation (3.30)
we have
k
E(U) = E[alE(ullci)+ •••+akE(~lci)]
= E[~atE(Utlci)]
Illustration 3.14. Assume two-stage sampling with simple random
sampling at both stages. Let X~t defined as follm~st be the estimator of
(3.31)
the population total:
lot m
Ni
x~ • -
r
mini
(3.32)
102
Exercise 3.17. Examine the estimator, x',
Equation (3.32).
Express
: it
·in other
forms that might help show its
logical
structure.
For example,
N
i
n
i
•Jor al fixed i what is --
r x
? Does it
seem like a reasonable way of
-:ni
j
ij
esti~ating
the populatio~
total?
To display x' as a linear
combination of random variables
it
is
convenient
to express
it
in the followin~ form:
M Nl
M N1
II Nm
H Nm
;x'
III'
[-
- Xl +••. + - - x
] +•.• + [- -- xml +... + - --- x
(3.33)
" ~\ ,,'~'
•.-',,
m n1. 1
m nl
lnl
m nm
m nrn
tnnm
i'
.F. ~ •., ~ ••
'
"'
I~. i'
I
:
•
I"
':' ~.i,',
:":1'"
-,
'.
"'?*~~',w:e
wan,t,::,.~o
::,find .the expecte.4.,yalue of x'
to determine whether
it
i
~
•
l'
,
.,
..
is ~qual
to the population
total.
Accordin~ to Theorem 3.8,
"
, .
.
,";..",
(3.34)
" •
~
..
,.~.
,
•• "
:I
•
,
}
~
+~
-.
-
\~.
-1
~•.~!.-
l { • ~~
.-Eq'4at,1,ons(3.34) and (3.35) are obtained simply by substituting
x' as
,
1,',
~.
_
i""-
t
tM .•..
random variable
in (3.30).
The ci now refers
to anyone of the m
'. ,
:pa~r~in the sample.
First we must solve
the conditional
expectation,
~ ....
,
'.
,
N
:,",E2(~:f i).
Since!!
and -.!. are constant with respect
to the conditional
"
'
,""
,
..
m
ni
:',
. "
. ~..
.
,.
, ..
,... ..
, ,ex~'o~a.Ci,on, and making use 0f Theorem 3. 7, we can ,,,rite
(3.35)
~
-,
I
,
'
.....
.. ',:'.
.. .
M m Ni ni
E2(x'li)
""- t -- t E2(xijli)
mini
j
( 3 • 36)
r"
'I ....
,
,1-
.
.- :'W~,~
fo.• an~ given psu in the sample that xij
is an element
in a
, si!llf>le' randcm saIlll'le from the psu and according
to Section 3.3 its
'r, ,
<; .:e~ect.ed value is
the psu mean, Xi••
That is,
103
and
Substituting the result from Equation (3.37) in Equation (3.36) gives
m
E2(x~li) = ~ E N i
m i
i i·
(3. 38)
Next we need to find the expected value of E2(x~li).
In Equation
i:.+"
.
-~
'."
.
•
#,"
':..~;..rj~··(
.<
'II:
•
#
#,
••
t, ~~'" : '-'~
"
"
,
, ,
random' ,- ,:..,'"
,
••••
ri~
"<:.}, '
'"
,
, ,
Mm.
urn
E [
~ X
]
- ~ EE (X
)
1 m'
i·
rn 1 i·
i
i
.•
',,"
·11
v,~.i',.",
variable which gives in lieu of Equation (3.38).
M m
E2(x~li) -;
~ Xi.
,
r.
'f,
;
I
~
From Theorem 3.,3
Therefore,
Since
(3.38), Ni is a random variable, as well as Xi.' associated with the first':
stage of sampling.
Accordingly, we will take Xi. - NiXi• as the
,
.
')'
m
E [M 'EX
]
1 m
i·
i
estimator of the population total.
M
Therefore, E(x~) • E Xi' - X ••
i
This shows that x~ is an unbiased
'",
'
- I
. ~
..,.
<:
....
-...
"
•... \.
~'
..
'
~..
~:,
3.10 CONDITIONAL VARIANCE
Conditional variance refers to the variance of a variable under a
specified condition or limitation.
It is related to conditional prob-
ability and to conditional expectation.
".!r1 •••.
\'
:.
l
t-
I~d.
I""'"
,~
~,
••.::
t~~~~:
:,,±~~,~'~~j'~r;
-,,'
.
, '.
(.~,
"
',,~..
104
To find the variance of x" (See Equation (3.32) or (3.33» the followin~
important theorem will be used:
:rheor~~~~.
The variance of x' is given by
where Vl is the variance for toe first stage of sampling and V2 is the
"conditional" variance for the second star,e.
We have discussed E2(x'li) and noted there is one value of E2(x'li)
for each psu in the population.
Hence VlE2(x'li) is simply the variance
of the M values of E2(X'!i).
In Theorem 3.9 the conditional variance, V2(X'!i) ,
V2(x'li) = E2{[x'-E~(x'li)]2
Ii}
by definition is
I,'
To understand V2(x'li) think of x' as a linear combination of random
variables (see Equation (3.33». Consider the variance of x' when i is
held constant.
All terms (random variables) in the linear cor.1bination
th
are now constant except those originating from samp1in~ within the i
psu.
Therefore, V2(x'li) is associated with variation amonR elements in
the ith psu.
V2(x" i) is a random variable wi th M values in the set, one
;-
for each psu.
Therefore, ElV2(x'l i) by definition is
H
= EP(i)V2(x'!i)
i
That is, E1V2(x'li) is an average 9f ~ values of V2(X'\i) weip,hted by
P(i),the probability that the ith psu had of bein~ in the sacple.
Three illustrations of the application of Theorem 3.9 will be given.
In each case there will be five steps in findinp, the variance of x':
Step 1, find E2(x'li)
Step 2, find V1E2(X"i)
105
Step 3, find v2(x"l i)
Step 4, find E1V2(x"ji)
Step 5, combine results from Steps 2·and 4.
Illust~~ti~~_!.15.
This is a simple illustration, selected because
we know what the answer is from previous discussion and a linear cornbina-
tion of random variables is not involved.
Suppose x" in Theorem 3.9 is
simply the random variable X where X.has an equal probability of being
anyone
of the Xij values in the set of N
variance of X,can be expressed as follows:
He know that the .
.-'
.
(3.39)
In the case of two-stage sampling an equivalent: method of selecting a
value of X is to select a psu first and then select an element within the
psu, the condition bein~ that P(ij) • P(i)p(jli) • ~.
This condition is
Ni
1
satisfied by letting P(i) - ~
and P(j/i) • ~.
We now want to find
i
VeX) by using Theorem 3.9 and check the result with Equation (3.39).
Step 1.
From the random selection specifications we know that
E2(X"'i) • Xi.
Therefore,
Step 2. VlE2(x"\i) - V1(Xi.)
We know that
equal to the
Ni
Xi. is a random variable that has a probability of ~
of being
ith value 1n the set Xl"."
~.
Therefore, by definition
of the variance of a random variable,
where
(3.40)
106
Step 3. By definition
Step 4. Since each value of V2(x'li) has a probability
M Ni Ni 1
-
2
= L N
L N- (Xij-Xi.)
i
j
i
(3.41)
Step 5. From,Equations (3.40) and (3.41) we obtain
The fact that Equations (3.42) and (3.39) are the same is verified
by Equation (1.10) in Chapter I.
(3.42)
Illustration 3.16. Find the variance of the estimator x' given by
Equation (3.3~) assuming simple random samp1inR at both stages of sampling.
Step 1. 'Theorem 3.7 is applicable.
That is,
mIli
_.'N •
E (x'i i) = H E
[::.!;
..-!. x
.1 if
2
ij
2 m ni
iJ
which means "sum the conditional expected values of each of the n terms
in Equation (3.33)."
With regard to anyone of the terms in Equation (3.33), the
conditional expectation is
E21: :: xijli] ~:
:: E2(xijli)
Therefore
H
= -m
N,
~ -
-x
n.. i·
~
M X.~.
:=---
m n.~
mn. u:
X.•
=
l:L~:~
~.
.,
m
n.
1J
1
(3.43)
With reference to Equation (3.43) and summinp with respect to j, we have
107
Hence Equation (3.43) becomes
m
M
- -
Step 2.
m
E Xi.
i
Find VIE2(x~li).
This is simple
m
EX
i
i
because --
t!\
in Equation
(3.44)
(3.44) is the mean of a random sample of m from the set of psu totals
Xl.' ••• ' ~.
Therefore,
where
(3.45 )
M
E(Xi -X.) 2
2
i
.
O'b,l
-,.
M
and
x.
In the subscript to 0'2, the "b" indicat,es between psu variance and "1"
distinguishes this variance from between psu variances in later illustra-
tions.
Step 3. Finding v2(x~li), is more involved because the conditional
variance of a linear combination of random variables must be derived.
However, this is analogous to using Theorem 3.5 for finding the variance
of a linear combination of random variables.
Theorem 3.5 applies except
that V(uli) replaces V(u) and conditional variance and conditional co-
variance replace the variances and covariances in the formula for V(u).
As the solution proceeds, notice that the strategy is to shape the problem
so previous results can be used.
Look at the estimator x~, Equation (3.33), and determine whether any
covariances exist.
An element selected from one psu is independent of an
I
7""
'~,
"••
I
108
element selected from another; but within a psu the situation is the same
as the one we had when finding the variance of the mean of a simple random
sample.
This suggests writing x~ in terms of xi. because the xi. 's are
independent.
AccordinR1y. we will start with
Hence
Since the x
's are independent
i·
I
,~,
' I
• ,
." •••
l~r •••
,
4
•• ~
"
~
•
-It
,
,
and since Ni is constant with regard to the"conditional variance
Since the sampling within each psu is simple random sampling
Ni-ni
2
V2(xi.li)-
ai
(N -1 ) ni
i
where
(3.46)
0.47)
Step 4. After substitutin~ the value of V2(xi.li) in Equation (3.40).
and then applying Theorem 3.3. we have
M2
Ni-ni
2,
E1V2(x~1i)
m
2
ai
-2
E El[Ni
N -1
-]
m
i
i
ni
Since the first staRe of saMplin~ was simple random sampling and each psu
had an equal chance of beinp, in the sample,
109
(1) at the first
N
i
- -
and (2)
N
Nt-nt
2
M
Nt-nt
2
2
<1i
1
E N2
O't
El[Ni
-]- -
N -1
n1
M
i i N -1 ni
i
1
Hence
M 2 Nt-ni
2
E v (x"'li)_ M
O'i
E Ni N -1
12m
i
i
ni
Step 5.
Combining Equation (3.48) and Equation (3.45)
2
M
2
2 M-m
O'b1
M
2 Ni-ni O'i
V (x"')- M -- --+-
L N
-
M-1
m
m
i i Ni-l
ni
I11ustr~~ion 3.17. ,The sampling specifications are:
stage select m psu's with replacement and probability P(i)
(3.48)
the answer is
(3.49)
\,
at the second stage a simple random sample of n elements is to be selected
froD.each of the m psu's selected at the first s~age.:"This will give a sam-
ple of n - milelements.
Find the variance of the sample estima,te of the
population total.
'The estimator needs to be changed because the p~u's are not selected
with equal probability.
Sample values need to be weighted by the recip-
rocals of their probabilities of selection if the estimator is to be
unbiased.
Let
P"'(ij) be the probability of element ij being in the sample,
th
P"'(i) be the relative frequency of the i
psu being in a sample
of m, and let
P"'(jli) equal the conditional probability of element ij being in
th
the sample given that the i
psu is already in the sample.
Then
N
i
According to the sampling specifications P"'(i) • m ~.
This prob-
ability was described as relative frequency because "probability of being
in a sample of m psu' s" is subject to misinterpretation.
th
The i
psu
110
can appear in a sanple nare than once and it is counted every time it
'th
appears.
That is, if the i
psu is selected more than once, a sanple of
-
~
n is selected within the i
psu every time that iltis selected.
By
Equation (3.50) means that every element has an equal probability of being
in the sample.
Consequently, the estiM<1tor is very simple,
substitution
P" (ij)
N
-
i
n
mn
n
=
[m U1 i:" ...N
•••N
~
(3.50)
N
x" --
mn
i(3.51)
I;
Exercise 3.18.
Show that.x", Equation (3.51), is an unbiased estimator
of the population total.
In finding V(x") our first step was to solve for E2(X'"i).
Step 1. By definition
Since i is constant with rep,ard to E2'
.•.
:, ~I
~
.. -
mn
(3.52)
Proceeding from Equation (3.52) to the foll~~ing result is left as an
exercise:
N
... -
m
(3.53)
Step 2. From Equation (3.53) we have
III
Since the X
's are independent
i·
Because the first stap,e of sampling is sanpling with probability propor-
tiona! to ~:i and \.rithreplac,ementt
Let
(3.54)
?
. N-
2
-
--
(J
m
b2
(3.55)
II:
lit Exerc:.ise3.19,. "1Prove ,that E(Xi)
•••X•• which sho\ls that it is
appropriate to use X.. in Equation (3.54).
Step 3. To find V2(x~li), first write the esti"~tor as
N
x~ = -
m
m
L Xi
i
.
(3.56)
(~k~~:~
'
~1:~:;';:
c .
Then, since the x
's are independent
i·
and
where
112
Therefore
1'•••••••
,"
"'~,'~
I'"
~
•
.0:>.
'.:.
~.
"'I.••
~
";~1~~~~i~
.;.~~~,~.;.
,
,'.:,J,f~
Step 4.
N2 1
--2 -
m
n
Since
the probability
of V2(x'li)
which becomes
N2 M, Ni ,.Ni-n 2
E1V2(x"1i)
-.-::
:I:,.:i- '.(N _l)oi
,
mn
i
i
Step 5. i Coimbining' EqUationl
1.( 3.55 r and Equation
(3.57) we have the
answer
(3.57)
(3.58)
.•.
.•~!.• ;: .•~~~
~'::
.,\>, ~~
r'
,
..
113
CHAPTER IV.
THE DISTRIBUTION
OF AN ESTIMATE
4.1
PRDPERnES
OF SIMPLE RANDOM SAMPLES
The distribution
of an estimate is a primary basis for judging the
accuracy of an estimate from a sample survey. But an estimate is only
one number. Howcan one numberhave a distribution?
Actually, "distri-
bution of an estimate" is a phrase that refers
to the distribution
of
all possible estimates that might occur under repetition of a prescribed
sampling plan and estimator (method.of estimation).
Thanks to theory
and empirical testing of the theory, .it
is not necessary to generate
physically
the distribution
of ,an estimate by selecting numeroussamples
,
,
anet"making an estimate from each. However, to have a tangible dist.ribu-
tion of an estimate as a basis for discussion, an illustration
has been
prepared.
Illustration
4.1.
Consider simple randomsamples of 4 from an
NI
81
assumedpopulation of 8 elements. There are nl (N-n)1 - 4T4T - 70 possible
samples.
In Table 4.1, the sample values for all of the 70 possible sam-
pIes of four are shown. The 70 samples were first
lis ted in an orderly
manner to facilitate
getting all of them accurately recorded. The mean,
i, for each sample was computedand the samples were then arrayed
according to the value of x for purposes of presentation
in Table 4.1.
The distribution
of x is the 70 values of x shownin Table 4.1,
including
the fact that each of the 70 values of x has an equal proDaDIlity or being
the estimate.
These 70 values have been arranged.as a frequency distribu-
tion in Table 4.2.
As discussed previously, one of the properties of simple random
sampling is that the sample average is an unbiased estimate of the popu-
lation average; that is, E(x) - X. This means that the distrIbution of
114
Table 4.l--Samples of four elements from a population of eight 1/
Sample
number
Values of
xi
x
..
------
2
8
.
S
1• Values of
amp e:
number:
Xi
.---
-x
2
s
..
----------
Ie
2
3
4
5
6
7
8
ges
10
lIs
12
13
14
15s
16
17
188
19s
20
21s
228
23s
248
25
268
278
28es
29
308
31s
328
338
34s
358
2,1,6,4
2,1,4,7
2,1,4,8
2,1,6,7
2,1,4,9
2,1,6,8
2,1,6,9
2,1,4,11
2,1,7,8
1,6,4,7
2,1.7,9 :
2,6,4,7
1,6,4,8
2,1,6,11
2,1,8,9
2,6,4,8
1,6,4,9
1,4,7,8
2,1,7,11
2,6,4,9
2,4,7,8
1,4,7,9
2,1,8,11
2,4,7,9
1,6,4,11
1,6,7,8
1,4,8,9
2,1,11,9
2,6,4,11
2,6,7,8
2,4,8,9
1,6,7,9
1,4,7,11
2,6,7,9
2,4,7,11
3.25
3.50
3.75
4.00
4.00
4.25
4.50
4.50
4.50
4.50
4.75 .
4.75
4.75
5.00
5.00
5.00
5.00
5.00
5.25
5.25
5.25
5.25
5.50
5.50
5.50
5.50
5.50
5.75
5.75
5.75
5.75
5.75
5.75
6.00
6.00
4.917
7.000
9.583
8.667
12.667
10.917
13.667
20.333
12.333
7.000
14.917
4.917
8.917 :
20.667 :
16.667
6.667
11.337
10.000 :
21. 583
8.917
7.583
12.250
23.000:
9.667
17.667
9.667
13.667
24.917
14.917
6.917
10.917
11.583
18.250
8.667
15.333
36s
378
38s
39s
40s
418
42
43es
44s
45s
46"
I
478
48s
49s
50s
51
52s
538
54
55
568
57
58
59
60s
61
62es
63
64
65
66
67
68
69
70e
1,6,8,9
1,4,8,11
2,6,8,9
2,4,8,11
1,6,7,11
1,4,11,9
1,7,8,9
6,4,7,8
2,6,7,11
2,4,11,9
2,7,8,'91'
1,6,8,.11
6,4,7,9
2,6,8,11
1,6,11,9
1,7,8,11
6,4,8,9
2,6,11;9
2,7,8,11
1,7,11,9
6,4,7,11
4,7,8,9
2,7,11,9
1,8,11,9
6,4,8,11
2,8,11,9
6,4,11,9
6,7,8,9
4,7,8,11
4,7,11,9
6,7,8,11
4,8,11,9
6,.7,11,9
6,8,11,9
7,8,11,9
6.00
6.00
6.25
6.25
6.25
6.25
6.25
6.25
6.50
6.50
6.150
6.50
6.50
6.75
6.75
6.75
6.75
7.00
7.00
7.00
7.00
7.00
7.25
7.25
7.25
7.50
7.50
7.50
7.50
7.75
8.00
8.00
8.25
8.50
8.75
12.667
19.333
9.583
16.250
16.917
20.917
12.917 .
2.917
13.667
17.667
9.667
17.667
4.333
14.250
18.917
17.583
4.917
15.333
14.000
18.667
8.667
4 .667 ;
14.917
18.917
8.917
15.000
9.667
1.667
8.333
8.917
4.667
8.667
4.917
4.333
2.917
1..1
X -
3
Values of X for
6, X4 - 4, X5 -
!(Xi-X/
N-1
= 12.
the population of eight e1e~nts
are Xl = 2, X2
7, X6 - 8, X7 - 11, X8 = 9; X = 6.00;
and
= 1,
x
Table 4.2--Sampling distribution of x
Relative frequency of i
Simple random
:Cluster sampling :Stratified random
sampling
sampling
:Illustration 4.1 .;Illustration 4.2 :Illustration 4.2
115
3.25
1
,
.
'"
3.50
1
~.~:~~~t:,,~~~t~
~
3.75
1
~v:~~
,' ....
'\ .......
r>;"
" 1,':I~!·/t~*:~/l
4.00
2
'~:~:>'·,.:(~ti~~~:
•••
4.25
1
4.50
4
4.75,
"
"
3
.,
.. '
,'.•J ~t:l.
i.
,)
5.00
5
.,,
5.25 ....
fi !~
i"1'-' 1iI11" .. 4
.
5.50
5
5.75
6
6.00
.~ •• ,i
."
4
6.25
6
6.50
5
6.75
4
7.00
5
7.25
3
,.
,
\.,?:~~~:
4-
7.50
4
\
: ~,';' ';
" ~
7.75
1
8.00
2
8.25
1
8.50
1
8.75
1
1
1
1
1
1
1
1
1
2
3
4
5
4
5
4
3
2
1
1
Total
Expected value
of i
Variance of x
70
6.00
1.50
6
6.00
3.29
36
6.00
0.49
--.--_0
0
_
116
X is centered on X.
If the theory is correct, the average of x for the
70 samples, which are equally likely to occur, should be equal to the
..
population average, 6.00.
The average of the 70 samples does equal 6.00.
From the theory of expected values, we also know that the variance
of x is given by
• 1.5
2
S~ • N-n L
x
N n
where
With reference do Illustration 4.1 and Table 4.1, S2 - 12.00 and S~ •
x
8-4 12
-8-"4
- 1.5.
The formula (4.1) can be verified by computing the
variance among the 70 values of i as follows:
.
2
' 2
2
(3.25-6.00) + (3.50-6.00) +••• + (8.75-6.00)
70
2
Since S is a population parameter, it is usually unknown.
Fortu-
nately, as discussed in Chapter 3, E(s2) - s2 where
(4.1)
2
s •
n
_ 2
t(xi-x)
i
n-l
I
-,I
1"
••
\
"
2
In Table 4.1, the value of s is shown for each of the 70 samples.
The
average of the 70 values of s2 is equal to s2.
The fact that E(s2) _ S2
is another important property of simple random samples.
used as an estimate of S2. That is t
2
N-n s2
s- - -- -
x
N n
is an unbiased estimate of the variance of x.
2
In practice s is
To recapitulate, we have just verified three important properties of
simple random samples:
117
(1) E(x). X
(2)
I¥-n
s-·
--
x
N
S
In
..
"
••
The standard error of x. namely S- • is a measure of how much i varies
x
under repeated sampling from X. Incidentally. notice that Equation (4.1)
shows how the variance of i is related to the size of the sample.
Now
we need to consider the form or shape of the distribution of x •
Definition 4.1.
The distribution of an estimate is often called the
sampling distribution.
It refers to the distribution of all possible
"I:
'
values of an';estimate that· could occur under a prescribed sampling plan.
""I
..
4.2
SHAPE OF THE SAMPLING DISTRIBUTION
For random sampling there is a large volume of literature on the
distribution of an estimate which we will not attempt to review.
In
practice. the distribution is generally accepted as being normal (See
f
I"~
'"
\'-<; "
,
.~,
,
--
"
Figure 4.1) unless the sample size is "small. II
The theory and empirical
tests show that the distribution of an estimate approaches the normal
distribution
rapidly as the size of the sample increases.
The closeness
of the distribution of an estimate to the normal distribution depends on:
(1) the distribution of X (i.e., the shape of the frequency distribution
of the values of X in the population being sampled). (2) the form of the
estimator.
(3) the sample design. and (4) the sample size.
It is not
possible to give a few simple. exact guidelines for deciding when the
degree of approximation is good enough.
In practice. it is generally a
matter of working as though the distribution of an estimate is normal but
being mindful of the possibility that the distribution might differ
ljj •.•••••••~ "~
I,.
t'
"J-o'lr,
'I'~
,),
"'~I
.r,...•
"
~
118
E(x"')-2a
..•
x
E(x"')-a
..•
x
E(x"')
,'.E (x "')+0
..•.,.: E (x "')+20
..•
...:'
. x'
.
x
Figure 4.l--Distribution
of an"estimate (normal distribution)
considerably from normal when the sample is very small and the population
distribution is highly skewed. 1/
It is very fortunate that the sampling distribution is approximately
normal as it gives a basis for probability statements about the precision
of an estimate.
As notation,x'" will be the general expression for any
estimate,and 0 ...
is the standard error of x ...•
x
Figure 4.1 is a graphical representation of the sampling distribution
of an estimate.
It is the normal distribution.
In the mathematical
equation for the normal distribution of a variable there are two parameters:
the average value of the variable, and the standard error of the variable.
1/ For a good discussion of the distribution of a sample estimate, see
Vol. I, Chapter l, Hansen, Hurwitz, and Madow.
Sample Survey Methods and
Theory,
John Wiley and Sons, 1953.
,
"
119
Suppose x~ is an estimate from a probability sample.
The characteristics
of the sampling distribution of Xl are specified by three things:
(1) the
expected value of Xl, E(xl),
which is the mean of the distribution;
(2) the
standard error of x#, a I' and (3) the assumption that the distribution
is
x
normal.
If x# is normally distributed, two-thirds of the values that Xl
99.7 percent of the estimates are within 3a
I from E(x#).
x
Exercise 4.1.
With reference to Illustration 4.1, find E(x) - a- and
------
x
E(x) + a-.
Refer to Table 4.2 and find the proportion of the 70 valuest~ ,;,~:
.
X
•• •.
could equal are between [E(xl)
- a I] and [E(x#) + a 1],95 percent of the
x
x
possible values of Xl are between [E(xl) - 2a I] and [E(xl) + 2a I]' and
x·
x
.~"
"
of x that are between E(x) - a- and E(x) + a-.
How does this compare with
x
x
the expected proportion assuming the sampling distribution of x is normal?
The normal approximation is not expected to be close, owing to the small
size of the population and of the sample.
Also compute E(x) - 20- and
x
E(x) + 2a- and find the proportion of the 70 values of i that are between
x
these two limi ts•
4.3
SAMPLE DESIGN
There are many methods of designing and selecting samples and of making
estimates from samples.
Each sampling method and estimator has a sampling
distribution.
Since the sampling distribution is assumed to be normal,
2
alternative methods are compared in terms of E(xl)
and a I (or 0 I).
x
x
For simple random sampling, we have seen, for a sample of nJ that
every possible combination of ~ elements has an equal chance of being the
sample selected.
Some of these pOSSible combinations (samples) are much
better than others.
It is possible to introduce restrictions in sampling
so some of the combinations cannot occur or so some combinations have a
>'
'
l'
,:~;~i~p"\
"
,,;
II
'\?_
120
higher probability of occurrence than others.
This can be done without
introducing bias in the ext~mate x~ and without losing a basis for esti-
mating a ~•. Discussion of particular sample designs is not a primary
x
purpose of this chapter.
Howevert a few simple illustrations will be
used to introduce the subject of design and to help develop concepts of
sampling variation.
Illustration 4.2.
Suppose the population of 8 elements used in
Table 4.1 is arranged so it consists of four sampling units as follows:
Sampling Unit
Elemen ts
Values of X
Sample Unit Total
1
lt2
Xl - 2t X - 1
3
2
2
3,4
X - 6t ,X4 - 4
10
3
3
5,6
X - 7, X - 8
15
5
6
4
7,S
X7 - llt Xs - 9
20
For sampling purposes the population now consists of four sampling
units rather than eight elements.
If we select a simple random sample of
two sampling units from the population of four sampling units, it is clear
that the sampling theory for simple random sampling applies.
This illus-
tration points out the importance of making a clear distinction between a
sampling unit and an element that a measurement pertains to. A sampling
unit corresponds to a random selection and it is the variation among sam-
pling units (random selections) that determines the sampling error of an
estimate.
When the sampling units are composed of more than one element t
the sampling is commonly referred to as cluster sampling because the ele-
ments in a sampling unit are usually close together geographically.
121
For a simple randomsample of 2 sampling units,
the variance of x ,
c
-
where x is the sample average per sampling unit,
is
c
where
S~
xc
. S2
N-n
c
-- -.
N
n
13.17
S2 • (3-12)2 + (10-12)2 + (15-12)2 + (20-12)2
158
N - 4, n • 2,. and
..•
-------------------------
.•..
-- -'-3
c
3
Instead of the average per sampling unit one will probably be interested
-x
-
c
in the average per element, which is x • 2 ' since there are two elements
in each sampling unit.
The variance of i is one-fourth of the variance
of xc. Hence, the variance of i is 13417 - 3.29.
There are only six possible, randomsamples as follows:
Sampleaverage per
2
Samplew SamplingUnits. I
sampling unit,
i
s
,
c
c-
1
1,2
6.5
24.5
2
1,3
9.0
72.0
3
1,4
11.5
144.5
4
2,3
12.5
12.5
5
2,4
15.0
50.0
6
3,4
17.5
12.5
where s2 •
c
and Xi is a sampling unit
total.
Be sure to notice
that s2 (which is the sainple estimate of S2) is the variance amongsampling
c
c
units in the sample, not--the variance amongindividual elements in the
sample. Fromthe list
of six samples, it
is easy to verify that i
is an
c
2
unbiased estimate of the population average per sampling unit and that Sc
is an unbiased estimate of 158 , the variance amongthe four sampling
3
122
units in the population.
Also. the variance amonR the six values of x is
13.17 which agrees with th~ formula.
The six possible cluster samples are among the 70 samples listed in
Table 4.1. Their sample numbers in Table 4.1 are 1. ~. 28. 43. 62. and
70. A "c" follows these sample numbers.
The sampling distribution for
the six samples is shown in Table 4.2 for comparison with simple random
sampling. 'It is clear fron inspection that random selection from these
six is less desirable ,than random selection from the 70. For ex,amp1e.
one of the two extreme averages. 3.25 or 8.75. has a
, "
u!
occurring for the cluster samp~ing and a probability
'II"
'I
~
I
~
,
1
probability of 3 of
1
of only 35 when
selecting a simple random sample of four elements.
In this illustration.
the sampling restriction (clusterinR" of elements) increaSed the sampling'
variance from 1.5 to 3.29.
It is of importance to note that the average variance among elements
within the four clusters is only 1.25.
(Students should compute the within
cluster variances and verify 1.25). This is much less than 12.0~ the
variance among the 8 elements of the population.
In ~ality.
the variance
among elements within clusters is usually less than the variance among all
elements in the population. because clusters (sampling units) are usually
composed of elements that are close together and elements that are close
together usually show a tendency to be alike.
Exercise 4.2.
In Illustration 4.2. if the average variance among
elements within clusters had been greater than l2.0~ the sampling variance
for cluster sampling would have been less than the sampling variance for a
simple random sample of elements.
Repeat what was done in Illustration 4.2
123
using as sampling units elements 1 and 6, 2 and 5, 3 and 8, and 4 and 7.
Study the results.
Illus~ation
4.3. Perhaps the most common method of sampling is to
assign sampling units of a population to groups called strata.
A simple
random sample is then selected from each stratum.
Suppose the population
used in Illustration 4.1 is divided into two strata as follows:
Stratum 1
Stratum 2
Xl c 2" X2 = 1, X3 • 6, X4 = 4
Xs • 7, X6 = 8, X7 = 11, X8 • 9
,
"j.:
<'
The sampling plan is to select a simple random sample of two elements
!
•
from each stratum.
There are 36 possible s~1es
of 4, two from each
"'
••.
I
stratum.
These 36 samples are identified in Table 4.1 by an s after the
sample number so you may compare the 36 possible stratified random samples;,
II(j· j
with the 70 simple random samples and with the six cluster samples.
Also,
; ",
see Table 4.2.
Consider the variance of x. We can write
x =
",-"
~
'.
~.
I..
where Xl is the sample average for stratum I and x2 is the average for
stratum 2. According to Theorem 3.5
S~ = (!.)(S~ + S~ + 2S- - )
x
4
Xl
x2
xlx2
We knm~ the covariance, S- - , is zero because the sampling from one
xlx2
stratum is independent of the sampling from the other stratum.
And,
since the sample within each stratum is a simple random sample,
52
Nl
-
2
2
:~Cnl
52
E (Xli-Xl.)
1
i
s- =
whe re
=
Xl
Nl
nl
1
N -1
1
124
The subscript "1" refers to stratum 1. s~ is of the same form as 5~ •
x2
Xl
Therefore ,
52
52
S~
I Nl-nl
N2-n2
1
-1.]
- - [ N
+
X
4
nl
·N
n2
I
2
Since
The variance, 0.49, is comparable to 1.5 in Illustration 4.1 and to 3.29 in
I
- '2' and nl - n2 - 2,
- 0.49
N2-n2
N2
52+52
[I
2]. 1[4.92+2.~2]
782
-
5~ • 1
x
8
Illustration 4.2.
In Illustration 4.2, the sampling units were groups of two elements and
the variance among these ~roups (sampling units) appeared in the formula
for the variance of x. In Illustration 4.3, each element was a.samp1ing
,
•
·,1'
unit but the selection process (randomization) was restricted to taking
one stratum (subset) at a time,so the sampling variance was determined by
variability within strata.
As you study sampling p1ans,form mental pictures
of the variation which the sampling error depends on. With experience and
\
accumulated knowledge of what the patterns of variation in various popula-
,:1:',,0<.
~.'~
tions are like, one can become expert in judging the efficiency of a1terna-
tive sampling plans in relation to specific objectives of a survey.
l
If the population ~d the samples in the above illustrations had been
larger, the distributions in Table 4.2 would have been approxinate1y nor-
mal.
Thus, since the form of the distribution of an estimate from a prob-
ability sample survey is accepted as being normal, only two attributes of
an estimate need to be evaluated, namely its expected value and its
variance.
'"I
'
125
In the above illustrations
ideal conditions were implicitly assumed.
Such conditions do not exist
in the real world so the theory must be
extended to fit,
'moreexactly, actual conditions.
There are numerous
sources of error or variation to be evaluated. The nature of the rela-
tionship between theory and practice is a major governing factor deter-
mining the rate of progress toward improvementof the accuracy of survey
resul ts.
Wewill nowextend error concepts towardmorepractical
settings.
4.4
RES·PONSE ERROR
So far,' we have discussed sampling under implicit assumptions that
measurements'are obtained from all n elements in a sample and that the
measurementfor each element is without error.
Neither ass:umptionfits,
exactly I, the'real world.
In addition,
there are "coverage" errors of
various kindS. For example, for a farm survey a farm is defined but
application of the definition
involves somedegree of ambiguity about
whether particular
enterprises satisfy
the definition.
Also, two persons
might have an interest
in the samefarm tract giving rise to the possibility
1...
'
that the tract might be counted twice (included as a part of two farms) or
omitted entirely.
Partly to emphasizethat error in an estimate is more than a matter
of sampling, statisticians
often classify
the numeroussources of error
into one of two general classes:
(1) Samplingerrors which are errors
associated with the fact that one has measurementsfor a s,ampleof elements
rather than measurementsfor all elements in the population, and (2) non-
sampling errors--errors
that occur whether sampling is involved or not.
Mathematical error models can be very complexwhenthey include a term for
'.r'"
:t;.
.
"
.',,~'~ •.
'",* "
•••••f ."")••••••••~..•.•
f ..~! '
.'
,"
126
each of many sources of error and atte1l1pt to represent
exactly
the real
world.
However, complicated error models are not always necessary,
depending upon the purposes.
For purposes of discussion,
two oversimplified
response-error models
will be used.
This will
introduce
the subject of responBt: error and give
some clues regarding
the nature of the impact of response error on the
distribution
of an estimate.
For simplicity,
we will
assume that a
measurement is obtained for each element in a random sample and that no
ambiguity exists
regarding
the identity
or definition
of an element.
Thus,
we will be considering
sampling error and re:sponse error simultaneously.
Illustration
4.4.
Let T1, ••• ,T
N
be the "true values" of some variable
for the N elements of a population •. The mention of true values raises
numerous questions
about what is a true value.
For example, what is your
:::11
true weight? Howwould you define
the true weight of an individual?
We
'·1 •. •·
will
refrain
from discussing
the problem of defining
true values and simply '-
assume that
true values do exist
according to some practical
definition.
Whenan attempt
is made to ascertain Ti, some value other
than Ti might
be obtained.
Call
the actual value obtained Xi'
The difference,
ei •
th
Xi - Ti,
is
the respon.se error
for
the i
element.
If
the characteristic,
for example, is a person's weight,
the observed weight, Xi'
for the i th
individual
depends upon when and how the measurement is
taken.
However,
for simplicity,
as~ume that Xi is always the value obtained regardless
of
the cortditions under which the measurement is taken'.
In other words,
th
assume that
the response error,
ei,
is constant
for the i
element.
In
this hypothetical
case, we are actually
sampling a population set of values
Xl"",XN
instead of a set of true values T1, ••• ,TN•
127
Under the conditions
as stated,
the sampling theory applies
exactly
to the set of population values Xl""
,~.
If a simple random sample of
elements is selected
and measurements for all
elements in the sample are
obtained,
then E(x) - X. That is,
if
the purpose is
to estimate
if
N
tTi
i---
N
".',,:
, ,
,~
:,~-~:
~I
•••
the estimate
is biased unless
if happens to' be equal to X. The bias
is
X - T which is appropriately
called
IIresponse bias. II
"
,
Then, the mean of a sblple :!random"sample;maybe 'expressed as
or, as
n
tX
i
X·--n
x-t+e
" i
.
'
,
,
'I'
"
;._
I" I
From the theory of expected values, we have
E(x) - E(t) + E(e)
Since E(x) - X and E(t) - T it
follows that
X - if + E(e)
Thus, x is a biased estimate of T unless E(e)- 0, where E(e)
N
tei
---
N
i.. ~-
That is, E(e). is
the average of the response errors,
ei,
for the whole
population.
For simple random sampling the variance of x is
S2
s_2 __N-n
X
where
x
N
n
N
t(X
i
-X)2
s2 __i
_
X
N-l
Howdoes the response error affect
the variance of X and of x? Wehave
th
already written
the observed value for
the i
element as being equal
to
128
its
true value plus a response error,
that
is, Xi • Ti + ei•
Assuming
random sampling, Ti and ei are random variables. We can use Theorem 3.5
from Chapter· III
and write
(4 •.3)
222
where Sx is
the variance of X, ST is
the variance of T, Se is
the response
variance
(that
is,
the variance of e),
and ST
is
the covariance of T and
. ,e
e.
The terms on the right-hand
side of Equat~on (4.3) cannot be evaluated
unless data on Xi and T1 are' available;
however,
the equation does show how
the response error
influences
the variance of X and hence of i,.
As a numerical example, assume. a population of five elements and the
following values
for T and X:
T1
~
ei
23
26
3
13
12
-1
17
23
6
25
25
0
7
9
2
,.
~~'~II'
.
'j"'
Average
17
19
2
Students may wish to verify
the following results,
especially
the variance
of e and the covariance of T and e:
~"
~I
,I
2
Sx • 62.5
S~ - 54.0
2
S - 7.5
e
S
• 0.5
T,e
As a verification
of Equation (4.3) we have
62.5 - 54.0 + 7.5 + (2)(0.5)
Fromdata in
129
n
_ 2
L(xi-x)
a simple randomsample one would computes~ • _i
_
x
n-l
"
2
and use N;n
:x as an estimate of the variance of x.
Is it clear that
2
2
2
Sx is an unbiased estimate of Sx rather than of ST and that the impact of
variation
in ei is included in s; ?
To summarize, response error cavsed a bias in x as an estimate of T
,
'.'
--
that was 'equal to X - T.
In addition,
it was a source of variation
included
'in the'standard error of x. To evaluate bias and variance attributable
to
response error,
information on Xi and Ti must be available.
Illustration
4~5.
In this' case, we assumethat the response error
where
for a given element is not constant.
That is,
if an element were measured
th
on several occasions, the observed values for the i
element could vary
even though the true value, Ti' remained unchanged. Let the error model be
Xij • Ti + eij
Xij is the observed value of X for the i th element whenthe
and
observation is taken on a particular occasion, j,
Ti is the true value of X for the i
th element,
eij
is the response error for the ith element on a particular
occasion, j.
Assume,for any given element, that the response error, eij,
is a random
variable. Wecan let e1j • ei + eij, where ei is the average value of eij
for a fixed i,
that is, ei - E(eijli).
This divides the response error
for the i th element into two components: a constant component,ei, and a
variable component,eij•
Bydefinition,
the expected value of eij
is zero
for any given element. That is, E(eijli)
• O.
130
Substituting ei + eij for e~j' the model becomes
(4.4)
..
:~
The model, Equation (4.4), is now in a good form for comparison with
the model in Illustration 4.4.
In Equation (4.4), ei, like ei in
Equation (4.2) is constant for a given element.
Thus, the two models
are alike except for the added term, eij, in Equation (4.4) which allows
th
for the possibility tbat the response error for the i
element might not
be constant.
"
I
I;':
1'1
Assume a simple random sample of n elements and one observation for
, , I
II
each element.
According to the model, Equation (4.4), we may now wri te
1',
I
1
'.
the sample mean as follows:
·'_1,1,
.,.
tti
te
~eij
i'
i i
. '.
x -- +
+
n
n
n
I
,
Summation with respect to j is not needed as there is only one observation
for each element in the sample.
Under the conditions specified the expected
value of x may be expressed as follows:
E(x) - T + e
where
N
tT
i
i
T --- N
-
ande
N
te
i
i---
N
The variance of x is complicated unless some further assumptions are
made.
Assume that all covariance terms are zero.
Also, assume that the
conditional variance of eij is constant for all values of i; that is, let
v(eijli) - 5;.
Then, the variance of x is
52
S~
s2
s~ _ N-n
....!. + N-n
.-!. +
.....!.
x
N
n
N
n
n
where
2
S- -
e
N - - 2
E (ei-e)
i
N-1
131
and s; is the conditional variance of eij, that is, V(eij1i).
For this
model the variance of x does not diminish to zero as n~N.
However, assuming
S2
e
N is large, the variance of x, which becomes N
,is probably negligible.
Definition 4.2.
Mean-Square Error.
In terms of the theory of expected
values the mean-square error of an estimate, x"", is E(x""-T)2 where T is the
target value, that is, the value, being estimated.
From the theory it is
easy to show that
r,
'~"j"
I 2
.1
,'1 4i' ,"
•.
,I 2'
- [E(x~)-Tj
"+ E[x"'-E(x"')l'
, .
,
'
Thus, the mean-square error, mse, can be expressed as follows:
where
B - E(x"")
- T
I
~:
(4.5)
(4.6)
and
2
C1
'"
x - E,[x""-E(x",)]2
(4.7)
'.
"
Definition 4.3.
Bias.
In Equation (4.5), B is the bias in x""as
an estimate of T.
. ~"i. ,
Definition 4.4.
Precision.
The precision of an estimate is the
~
standard error of the estimate, namely,
C1
"" in Equation (4.7).
x
Precision is a measure of repeatability.
Conceptually, it is a
measure of the dispersion of estimates that would be generated by repetition
of the same sampling and estimation procedures many times under the same
conditions.
With reference to the sampling distribution, it is a measure
of the dispersion of the estimates from the center of the distribution and
, ~,.~.
':'
".", , ,,t
"~\' .'.~.
I' ~~I"'·~
'·I~·
'
;:;,.",..Itj,~r;.k;.
,
"/"
',~ ~:~
,.,
'\'
":
.,,')
"
132
does not include any indication of where the center of the distribution
is in relation
to a target.
In Illustrations
4.1, 4.2, and 4.3,
the target value was implicitly
assumedto l,>eX;
that is, T was equal to X. Therefore, Bwas zero and
the mean-square error of x'" was the same as the variance of x...• In
Illustrations
4.4 and 4.5 the picture was broadened somewhatby intra-
ducing respon.se error and examining, theoretically,
the impact of response
error on E(x"') and a .••
In practice manyfactors have potential
for
x
influencing the sampling distribution of x...• That is,
the data in a
sample are subject
to error that might be attributed
to several sources.
Fromsample data an estimate, x", is computedand an estimate of the
variance of x" is also computed. Howdoes one interpret
the results?
In
Illustrations
4.4 and 4.5 we found that response error could be divided
into bias and variance.
The error from any source can, at least concep-
tually, be divided into bias and variance. Anestimate from a sample is
subject
to the combinedinfluence of bias and variance corresponding to
each of the several sources of error.
'Whenan estimate of the variance
of x'" is computedfrom sample data,
the estimate is a comination of
variances that might be identified with various sources. Likewise the
difference between E(x") and T is a combination of biases that might be
identified with various sources.
Figure 4.2 illustrates
the sampling distribution of x'" for four
different
cases: A, no bias and low standard error,; B, no bias and larg,e
standard error; C, large bias and low standard error; and D, large bias
and large standard error.
The accuracy of an estimator is sometimesdefined
as the square root of the mean-squareerror of the estimator.
According
133
E(x")
1 I
T
I
T
E(x'")
B: No bias--large
standard error·
I
T
..1
"
T
E(x"}
A:
No bias--low standard error
"
I'
",
!
••..
':',
c: Large bias--low standard error
D: Large bias-large
standard error
Figure 4.2--Examples of four sampling distributions
T-- -"
". E(x")
Figure 4.3--Sampling distribution--
Each small dot corresponds to an estimate
• t
;.':
~
,
.
134
to that definition,
we could describe estimators
having the four sampling
distributions
in Figure 4.2 as follows:
In case A the estimator
is precise
and accurate;
in B the estimator
lacks precision
and is therefore
inaccurate;
in C the estimator
is precise but inaccurate
because of bias,
and in D-ehe
estimator
is
inaccurate
because of bias and low precision.
Unfortunately,
it
is generally not possible
to determine,
exactly,
the magnitude of bias
in an estimate,
or of a particular
component of bias.
However, evidence of the magnitude of bi~s is often available
from general
experience,
from knowledge of howwell
the survey processes were performed,
and from special
investigations.
The author accepts a point of view that
the mean-sq';lare error
is an appropriate
concept of accuracy to follow.
In
that context,
the concern becomes a matter 0 f the magnitude 0f the mse and
the ,size of B re1at:ive to ax" •. ::.:rhat,..viewpoint is
important because it
is
not possible
to bel certain
that B is zero.
Our goal should be to prepare
survey specifications
and to conduct survey operations
so B is small
in
relation
to a .•• Or, one might say we want the mse to be minimumfor a
x
given cost of doing the survey. Waysof getting evidence on the magnitude
of bias
is a major subject
and is outside
the scope of this publication.
As indicated
in the previous paragraph,
it
is
important
to know some-
thing about the magnitude of the bias, B, relative
to the standard error,
ax".
The standard error
is controlled
primarily by the design of a sample
and its
size.
For many survey populations,
as the size of the sample
increases,
the standard error becomes small relative
to the bias.
In fact,
the bias might be larger
than the standard error even for samples of
moderate size,
for example a few hundred cases, depending upon the circ~
stances.
The point
is
that
if
the mean-square error
is
to be small,
both
135
Band c ~ must be small.
The approaches for reducing B are very different
x
from the approaches for reducing c ..•• The greater
concern about non-
x
sampling'error
is bias rather
than impact on variance.
In the design and
selection
of samples and in the processes of doing the survey an effort
is
made to prevent biases
that are IIsamplingll in origin.
However, in survey
work one must be constantly
aware of potential
biases and on the alert
to
minimize biases
as well as random error
(that
is, c ...).
x
The above discussion puts a census in the same light
as a sample.
Results
from both have a mean-square error.
Both are surveys with refer-
ence to use olf results;
Uncertain
inferences
are involved in the use of
results
from a census as well as 'from--a sample.
The only difference
is
that
in' a census' one attempts
to get a measurement for all N elements,
I
'"
.~"
~lo
t
,
but making n • N'does no't reduce the mse to zero.
Indeed, as the sample
"
'
size
increas'es',
'there
is no positive
assurance
that
the mse will
always
decrease;
because,
as the variance component of the mse decreases,
the
bias component might increase.
This can occur especially when the popu-
lation
is
large and items on the questionnaire
are such that simple,
accurate
answers are difficult
to obtain.
For a large sample o,r a census,
compared to a small sample,
it might be roore difficult
to control
factors
that cause bias.
Thus, it
is possible
for a census to be less accurate
(have a larger mse) than a sample wherein the sources of error
are more
adequately controlled.
Muchdepends upon the kind of information being
collected.
4.5
BIAS AND STANDARD ERROR
The words IIbias,"
"biased, II and "unbiasedll have a wide variety
of
meaning amongvarious
individuals.
As a result,
much confusion exists,
,,~
'\
';.~
!"}:\;~>~"
" I'
'. ~ '
:f'
':},
'J,'-
136
espec~ally since the terms are often used loosely.
Technically,
it
seems
logical
to define the bias
in an estimate as being equal to B in Equation
(4.6), which is the difference between the expected value of an estimate
and the target value.
But, except for hypothetical
cases, numerical values
do not exist
for either E(x') or the target T. Hence, defining an unbiased
estimate as one where B - E(x')
- T - 0 is of little,
if any, practical
value unless on&is willing
to accept the target
as being equal to E(x').
From a saDipling point of view there are conditions
that give a rational
basis
for accepting E(x') as the target.
However, regardless
of how the
, ,I'
'I
.
target
is defined,
a good practical
interpretation
of E(x')
is needed.
,
I,
'
It has become commonpractice
among-surv~y statisticians
to -call an
estimate unbiased when it
is based on methods of sampling and estimation
f'
'
'II.
"
that are "unbiased."
For example, in Illustration
4.4, x would be referred
:
;
~'I ~
:
i
1,
to as an unbiased estimate--unbiased
because the method of slampling and
estimation was unbiased.
In other words, since x was an unbiased estimate
of i, i could be interpreted
as an unbiased estimate of the result
that
would have been obtained if all elements in the population had been
measured.
In Illustration
4.5 the expected value of x is more difficult
to
describe.
Nevertheless, with reference
to the method of sampling and
estimation,
i was "unbiased" and could be called an unbiased estimate
even though E(x) is not equal to T.
The point 1s that a simple statement which says, "the estimate
is
unbiased" is incomplete and can be very misleading,
especially
if one is
not familiar with the context and concepts of bias.
Calling an estimate
unbiased is equivalent
to saying the estimate
is an unbiased estimate of
~ .
,
"
•,
... , ..
-,
: \:,~.~
'
,
137
its expected value. Regardless of how"bias" is defined or used, E(x"')
is the meanof the sampli~g distribution of X; and this concept of E(x')
is very important because E(x') appears in the standard error, a ..•, of x'"
x
as well as in B. See Equations (4.6) and (4.7).
As a simple concept or picture of the error of an estimate from a
survey, the writer
likes the analogy between an estimate and a shot at
a target with a gun or an arrow. Think of a survey being replicated
manytimes using the samesampling plan, but a different
sample for each
replication.
Each replication would provide an estimate that corresponds
to a shot at a target.
'I
lC
In Figure 4.3, each dot corresponds to an estimate from one of the
,
'
replicated samples. The center of the cluster of dots is labeled E(x')
because' it corresponds to the expected value of an estimate.
Aroundthe
point lex')
a circle
is drawnwhich contains two-thirds of the points.
The radius of this circle corresponds to a ..•, the standard error of the
x
estimate.
The outer circle has a radius of two standard errors and con-
tains 9S percent of the points.
The target
is labeled T. The distance
.
'
between T and Eex"') is bias, which in the figure is greater
than the
standard error •
In practice, we usually have only one estimate, x"', and an estimate,
s ..•, of the standard error of x..•• With reference to Figure 4.3,
this
x
1II88lU1 one point and an estimate of the radius of the circle around E(x"')
that would contain two-thirds of the estimates in repeated samplings. We
do not knowthe value of E(x"'); that is, we do not knowwhere the center
of the circles
is.
However,whenwe makea statement about the standard
error of x"', we are expressing a degree of confidence about howclose a
1.1
138
particular estimate prepared from a survey is to E(x'); that is, how
close one of the points in Figure 4.3 probably is to the unknown point
E(x').
A judgment as to how far E(x') is from T is a matter of how T
is defined and assessment of the magnitude of biases associated with
various sources of error.
Unfortunately, it is not easy to make a short, rigorous, and complete
interpretative statement about the standard error of x'.
If the estimated
standard error of x' is three percent, one could simply state that fact
and not make an interpretation.
It does not help much to say, for example,
that the odds are about two out of three that the estimate is within three
percent of its expected value, because a person familiar with the concepts
already understands that and it probably does not help the person who is
.l
unfamiliar with the concepts.
Suppose one states, "the standard error of
x' means the odds are two out of three that the estimate is within three
percent of the value that would have been obtained from a census taken
under identically the same conditions."
That is a good type of statement
to make but, when one engages in considerations of the finer points,
interpretation of "a census taken under identically the same conditions"
is needed--especially
since it is not possible to take a census under
identically the same conditions.
In summary, think of a survey as a fully defined system or process
including all details that could affect an estimate, including:
the method
of sampling; the method of estimation; the wording ,of questions; the order
of the questions on the questionnaire; interviewing procedures; selection,
training, and supervision of interviewers; and editing and processing of
•
\
139
data.
Conceptually,
the sampling is then replicated many times, holding
all
specifications
and conditions constant.
This would generate a sam-
pIing dis~ribution
as illUstrated
in Figures 4.2 or 4.3. Weneed to
recognize that a change in any-of the survey specifications
or conditions,
regardless
of how trivial
the change might seem, has a potential
for
changi.ng the sampling distribution,
especially
the expected value of
.•
x •
,-
,
r
I
t'",
I"r~I
Changes in survey plans, even though the definition
of the parameters
being estimated remains unchanged, often result
in discrepancies
that
are larger
than the random error
that can be attributed
to sampling.
The points discussed in the latter
part of this chapter were included
to emphasize that muchmore than. a well designed sample is required
to
assure accurate
results.
Goodsurvey planning and managementcalls
for
-"f1~1'!' I
•
'.'1, I'
.
'. evaluation of errors
from all
sources and for trying
to balance the effort
to control error
from various sources so the mean-square error will be
within acceptable
limits
as economically as possible.
;:"vf:~~-
•••,,#,
\.
I